CHAPTER VIII
DIFFERENTIAL CALCULUS
The subject matter of this chapter is nothing else but the elementary
theorems of calculus, which however are presented in a way which will
probably be new to most students. That presentation, which throughout
adheres strictly to our general "geometric" outlook on analysis, .aims at
keeping as close as possible to the fundamental idea of calculus, namely the
"local" approximation of functions by linear functions. In the classical
teaching of calculus, this idea is immediately obscured by the accidental
fact that, on a one-dimensional vector space, there is a one-to-one cor-
respondence between linear forms and numbers, and therefore the deriv-
ative at a point is defined as a number instead of a linear form. This slavish
subservience to the shibboleth of numerical interpretation at any cost
becomes much worse when dealing with functions of several variables:
one thus arrives, for instance, at the classical formula (8.9.2) giving the
partial derivatives of a composite function, which has lost any trace of
intuitive meaning, whereas the natural statement of the theorem is of course
that the (total) derivative of a composite function is the composite of their
derivatives (8.2.1), a very sensible formulation when one thinks in terms of
linear approximations.
This " intrinsic" formulation of calculus, due to its greater " abstraction,"
and in particular to the fact that again and again, one has to leave the initial
spaces and to climb higher and higher to new "function spaces" (especially
when dealing with the theory of higher derivatives), certainly requires some
mental effort, contrasting with the comfortable routine of the classical
formulas. But we believe that the result is well worth the labor, as it will
prepare the student to the still more general idea of calculus on a differentiable
manifold, which we shall develop in Chapters XVI to XX. Of course, he will
observe that in these applications, all the vector spaces which intervene have